If f(2)=6 and f'(x)=sinx^3, find f(4)

[Slithus]

The answer I'm supposed to get is 5.986, but I'm not sure how to get it. I think it involves using an integral from 2 to 4. Can anyone explain to me how I am supposed to solve this?

 

[Slithus]

The answer I'm supposed to get is 5.986, but I'm not sure how to get it. I think it involves using an integral from 2 to 4. Can anyone explain to me how I am supposed to solve this?

Problem is in the title: If f(2)=6 and f'(x)=sinx^3, find f(4)

 

[Dr.Peterson]

The answer I'm supposed to get is 5.986, but I'm not sure how to get it. I think it involves using an integral from 2 to 4. Can anyone explain to me how I am supposed to solve this?

You could do that integral, though there will be another step. If I were you, I would try what you have in mind, and then check to see if my answer satisfied the conditions of the problem. Then you can adjust your thinking if it doesn't work out -- it's a great way to really learn.

The other approach would be to find an antiderivative (indefinite integral), and determine the constant of integration from the condition given. Again, check your answer when you finish.

On the other hand, assuming the function is sin(x^3), you won't be able to integrate it. Might this be a problem in numerical integration? What is the context?

 

[BigBeachBanana]

Rewrite the integral as follow:
$$\int \sin^3x\,dx=\int \sin x(1-\cos^2x)\,dx=\int \sin x\,dx - \int \sin x\cos^2x\,dx$$Use a u-subsitution for the second integral.

 

[Dr.Peterson]

Rewrite the integral as follow:
$$\int \sin^3x\,dx=\int \sin x(1-\cos^2x)\,dx=\int \sin x\,dx - \int \sin x\cos^2x\,dx$$Use a u-subsitution for the second integral.

That assumes that the integrand is (sin(x))^3, rather than sin(x^3). But Wolfram Alpha gives the expected answer, 5.986, when I assume the latter, which is the proper way to interpret what was written in the OP.

 

[blamocur]

The answer I'm supposed to get is 5.986, but I'm not sure how to get it. I think it involves using an integral from 2 to 4. Can anyone explain to me how I am supposed to solve this?

I am getting 5.9931, and I am pretty sure that I am better than Wolfram Alpha: I've just asked the latter to integrate $x^3$ for the same interval and got 59.76